raoult's law and aerosol compositon

Raoult's Law

The vapor pressure and composition in equilibrium with a solution can yield valuable

information regarding the thermodynamic properties of the liquids involved. Raoult’s law

relates the vapor pressure of components to the composition of the solution. The law

assumes ideal behavior. It gives a simple picture of the situation just as the ideal gas law

does. The ideal gas law is very useful as a limiting law. As the interactive forces between

molecules and the volume of the molecules approache zero, so the behavior of gases

approach the behavior of the ideal gas. Raoult’s law is similar in that it assumes that the

physical properties of the components are identical. The more similar the components the

more their behavior approaches that described by Raoult’s law.

Using the example of a solution of two liquids, A and B, if no other gases are present the

total vapor pressure Ptot above the solution is equal to the sum of the vapor pressures of the

two components, PA and PB.

If the two components are very similar, or in the limiting case, differ only in isotopic

content, then the vapor pressure of each component will be equal to the vapor pressure of

the pure substance Po times the mole fraction in the solution. This is Raoult’s law.

Thus the total pressure above solution of A and B would be

Graphically this can be represented by a diagram in which the horizontal axis gives the

composition from one pure component to the other as shown below.

With actual mixtures the plots of vapor pressure vs. composition usually depart from the

straight lines shown above, some curve above, some below due to intermolecular forces.

Below is an example of a positive deviation from ideality. The thin lines represent the ideal

behavior.

As the curves approach the extremes of the pure components, molecules of the minor

component are surrounded by molecules of the major component. Thus the departures from

the ideal give information about the interaction of the components. The partial vapor

pressure above the solution is obviously related to the escaping tendency and thus the

chemical potential. At equilibrium the chemical potential of a component i (µi) is the same in

the solution and in the vapor phase.

Koo et al., 2003

Some thoughts & derivations….

Ptot = PA0xA + PB

0xB Raoult’s Law

cieq

= xi,kci*

where xi,k is mole fraction of species i in section k; ci* is effective saturation

concentration of species i

ct,i = cg,i + ca,i = αi ∆HCi Mass balance

ca,i = ct,i – xici* for i = 1,n

ci = m/V = PiMi/(RT) Ideal gas law

total secondary organic mass in aerosol = Ca = Σca,i = Σct,i – Σxici*

,

,0

0

a i

ii

a j

j j

c

Mx

cc

M M

=

+∑ definition of mole fraction, but what about water, inorganics?

, ,

,

, ,0 0

0 0

a i a i i

ii ia j j i j j a j

a j a jj i j j

j jj j

c c c

M MC HC c HC c

c cc c

M M M M

α α

∗

∗= ∆ − ⋅ = ∆ − =

+ +

∑∑ ∑ ∑ ∑

∑ ∑

,

,

,0

0

a i i i

i ia j j a j

a jj j

j j

c P M

MC HC c

cc

M M

α

∗

= ∆ − =

+

∑∑ ∑

∑

,

,

,0

0

a i i

ia j j a j

a jj j

j j

c P

C HC ccc

M M

α

∗

= ∆ − =

+

∑∑ ∑

∑

Define weighted average molecular weight of organic species condensed onto aerosol:

,

,

a j j

j

a k

k

c M

Mc

=∑

∑

Then can approximate mole fraction using this:

,

,

, 0,0

0

0

a i

a i

i

a j

j a j

j

ccMx

c c Mcc

MM M

≈ =

++∑ ∑

Leading to:

,

,

0,

0

a i i

ia j j a j

j j

a j

j

M c P

C HC cc M

cM

α

∗

= ∆ − =

+

∑∑ ∑

∑

Some limiting cases:

(A) all Pi* values equal 1 (atmosphere)

,

,

,

0

0

a i

ia j j a j

j ja j

j

c

C HC cc

c

M M

α= ∆ − =

+

∑∑ ∑

∑

,

, ,

, ,

0 0

0 0

11

a i

ij j a j a j

j j ja j a j

j j

c

HC c cc c

c c

M MM M

α

∆ = + = + + +

∑∑ ∑ ∑

∑ ∑

Thus,

,

,

0

0

11

j j

j

a a j

j

a j

j

HC

C c

cc

M M

α ∆

= = + +

∑∑

∑

(B) all Pi* values equal 0

,a j j a j

j j

C HC cα= ∆ =∑ ∑

(C) all Pi* values the same = P

*

0,

0

1

j j

j

a

a j

j

HC

C

P M

c Mc

M

α

∗

∆

= + +

∑

∑

Note that in these cases, Ca is in the denominator of the right hand side. If we rearrange and

solve for Ca (for case (C)):

0

0

0

1

11

a

o

j j

j

CM

Pc MM

HC c Mα

∗

= + + − ∆ ∑

Just for simplicity, assume M0 = M =M

0 0

0

1

11

1

a

j j

j

CMP

c c

HC cα

∗= + + − ∆

∑

(1) Should denominator include everything (not just organic) in the aerosol phase?

(2) Note the ci* is “effective” saturation concentration, which includes the effects of processes

like oligimer formation, acid-base equilibria, condensed phase oxidation, condensed

phase activity, …; and thus, in general ci* ≤ ci, and it could much less!

Let’s take case (C) and add water to the aerosol. cw is concentration of water in the aerosol

phase. Thus (with assumptions Pi*=P*

, and 0

M M M= = ),

,

,

,00 ,

01818

a i

a iii

a j wwa j

jj j

c

cMx

c c Mc cc c

M M

= =

+ ++ + ∑∑

,

,

0 ,18

a i

ia j j a j

wj ja j

j

P M c

C HC cc M

c c

α

∗

= ∆ − =

+ +

∑∑ ∑

∑

Leading to:

2

0 0

0

0

2 2

0 0

018 18

1

18

18

4 4

2 2

18

w wa a j j j j

j j

wj j

j

wj j

j

a

wj j

j

c M c MC C c P M HC HC c

l

c Mm c P M HC

c Mn HC c

m m nl m m nC

l

c Mc P M HC c

α α

α

α

α

∗

∗

∗

+ + + − ∆ − ∆ + =

=

= + + − ∆

=− ∆ +

− ± − − + −= =

− + + − ∆ + + =

∑ ∑

∑

∑

∑2

04

18 18

2

w wj j j j

j j

c M c MP M HC HC cα α∗

+ − ∆ + ∆ + ∑ ∑

2

- 1/2 co - 1/36 cw M - 1/2 Pstar M + 1/2 aHC + 1/36 (324 co + 36 co cw M

2 2 2

+ 648 co Pstar M + 648 co aHC + cw M + 36 cw M Pstar + 36 cw M aHC

2 2 2 1/2

+ 324 Pstar M - 648 Pstar M aHC + 324 aHC )

>>

2

0 0

2 2 2 2 20

2

324 36 648 648

136 36 324

2 36 2 2 36

648 324

o w o i i

ii i

w ia w w w i i

i

i i i i

i i

c c c M c P M c HCHC

c c M P MC c M c P M c M HC P M

P M HC HC

α

α

α

α α

∗

∗∗ ∗

∗

+ + + ∆ ∆ =− − − + + + + + ∆ + − ∆ + ∆

∑∑

∑

∑ ∑

( ) ( )2

0 0

0

2

2

18 36 181

2 2 36 2 3618

w w i ii i

ii w

a

i i

i

c Mc c Mc P M HCHCc c M P M

C

P M HC

αα

α

∗

∗

∗

+ + + + ∆ +∆ = − + + + − ∆

∑∑

∑

This equation is display graphically on the following plots. Parameters given in the plot titles.

Raoult's Law and Mass Balance Calculation

cw = co, Pstar=1e-4, M=200

1.00E-05

1.00E-04

1.00E-03

1.00E-02

1.00E-01

1.00E+00

1.00E-06 1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02

co

Fra

cti

on

in

aero

so

l p

has

e

1.00E-06 2.15E-06 4.64E-06 1.00E-05 2.15E-05 4.64E-05 1.00E-04 2.15E-04 4.64E-04

1.00E-03 2.15E-03 4.64E-03 1.00E-02 2.15E-02 4.64E-02 1.00E-01 2.15E-01 4.64E-01

1.00E+00 2.15E+00 4.64E+00 1.00E+01 2.15E+01 4.64E+01 1.00E+02

a*HC=


cw = 0, Pstar=1e-4, M=200

1.00E-05

1.00E-04

1.00E-03

1.00E-02

1.00E-01

1.00E+00

1.00E-06 1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02

co

Fra

cti

on

in

aero

so

l p

ha

se

1.00E-06 2.15E-06 4.64E-06 1.00E-05 2.15E-05 4.64E-05 1.00E-04 2.15E-04 4.64E-04

1.00E-03 2.15E-03 4.64E-03 1.00E-02 2.15E-02 4.64E-02 1.00E-01 2.15E-01 4.64E-01

1.00E+00 2.15E+00 4.64E+00 1.00E+01 2.15E+01 4.64E+01 1.00E+02

a*HC=


cw = 10 x co, Pstar=1e-4, M=200

1.00E-05

1.00E-04

1.00E-03

1.00E-02

1.00E-01

1.00E+00

1.00E-06 1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02

co

Fra

cti

on

in

ae

ros

ol p

hase

1.00E-06 2.15E-06 4.64E-06 1.00E-05 2.15E-05 4.64E-05 1.00E-04 2.15E-04 4.64E-04

1.00E-03 2.15E-03 4.64E-03 1.00E-02 2.15E-02 4.64E-02 1.00E-01 2.15E-01 4.64E-01

1.00E+00 2.15E+00 4.64E+00 1.00E+01 2.15E+01 4.64E+01 1.00E+02

a*HC=



1.00E-05

1.00E-04

1.00E-03

1.00E-02

1.00E-01

1.00E+00

1.00E-06 1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02

co

Fra

cti

on

in

ae

ros

ol p

ha

se

1.00E-06 2.15E-06 4.64E-06 1.00E-05 2.15E-05 4.64E-05 1.00E-04 2.15E-04 4.64E-04

1.00E-03 2.15E-03 4.64E-03 1.00E-02 2.15E-02 4.64E-02 1.00E-01 2.15E-01 4.64E-01

1.00E+00 2.15E+00 4.64E+00 1.00E+01 2.15E+01 4.64E+01 1.00E+02

a*HC=



1.00E-05

1.00E-04

1.00E-03

1.00E-02

1.00E-01

1.00E+00

1.00E-06 1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02

co

Fra

cti

on

in

ae

ros

ol p

hase

1.00E-06 2.15E-06 4.64E-06 1.00E-05 2.15E-05 4.64E-05 1.00E-04 2.15E-04 4.64E-04

1.00E-03 2.15E-03 4.64E-03 1.00E-02 2.15E-02 4.64E-02 1.00E-01 2.15E-01 4.64E-01

1.00E+00 2.15E+00 4.64E+00 1.00E+01 2.15E+01 4.64E+01 1.00E+02

a*HC=



1.00E-05

1.00E-04

1.00E-03

1.00E-02

1.00E-01

1.00E+00

1.00E-06 1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02

co

Fra

cti

on

in

aero

so

l p

has

e

1.00E-06 2.15E-06 4.64E-06 1.00E-05 2.15E-05 4.64E-05 1.00E-04 2.15E-04 4.64E-04

1.00E-03 2.15E-03 4.64E-03 1.00E-02 2.15E-02 4.64E-02 1.00E-01 2.15E-01 4.64E-01

1.00E+00 2.15E+00 4.64E+00 1.00E+01 2.15E+01 4.64E+01 1.00E+02

a*HC=



1.00E-05

1.00E-04

1.00E-03

1.00E-02

1.00E-01

1.00E+00

1.00E-06 1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02

co

Fra

cti

on

in

ae

ros

ol p

hase

1.00E-06 2.15E-06 4.64E-06 1.00E-05 2.15E-05 4.64E-05 1.00E-04 2.15E-04 4.64E-04

1.00E-03 2.15E-03 4.64E-03 1.00E-02 2.15E-02 4.64E-02 1.00E-01 2.15E-01 4.64E-01

1.00E+00 2.15E+00 4.64E+00 1.00E+01 2.15E+01 4.64E+01 1.00E+02

a*HC=

raoult's law and aerosol compositon

Documents